Abstract
This article studies several properties of the half-space last passage percolation, in particular the two-time covariance. We show that, when the two end-points are at small macroscopic distance, then the first-order correction to the covariance for the point-to-point model is the same as the one of the stationary model. In order to obtain the result, we first derive comparison inequalities of the last passage increments for different models. This is used to prove tightness of the point-to-point process as well as localization of the geodesics. Unlike for the full-space case, for half-space we have to overcome the difficulty that the point-to-point model in half-space with generic start and end-points is not known.
Funding Statement
The work of P.L. Ferrari was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—GZ 2047/1, projekt-id 390685813 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Projektnummer 211504053—SFB 1060.
The work of A. Occelli was supported in part by ERC-2019-ADG Project 884584 LDRam, and was partially developed while A.O. was a postdoctoral fellow at MSRI during the Program “Universality and Integrability in Random Matrix Theory and Interacting Particle Systems”.
Acknowledgments
The authors are grateful to J. Baik for the detailed explanation on how to setup the correct Riemann–Hilbert problem and to M. Duits, T. Krieckerbauer and T. Bochner for various discussions on the Riemann–Hilbert techniques, and to G. Barraquand for exchanges concerning their work on half-space LPP.
Citation
Patrik Ferrari. Alessandra Occelli. "Time-time covariance for last passage percolation in half-space." Ann. Appl. Probab. 34 (1A) 627 - 674, February 2024. https://doi.org/10.1214/23-AAP1974
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